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11637 Foundations of Computational Data Science > Linear Algebra with NumPy Primer > Flashcards

Linear Algebra with NumPy Primer Flashcards

1
Q

What does this mean?

A

the set of all real-valued matrices with n rows and d columns

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2
Q

What is this called? What does it mean?

1(x)

A
  • binary indicator function,
  • returns 1 if x is a true boolean and 0 otherwise
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3
Q

What does this mean?

A

the set of all real-valued n-dimensional vectors

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4
Q

What does this mean?

A

the set of real numbers

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5
Q

What is this?

A
  • A pair of input vector and output scalar
  • The superscript indicates the index of this pair, not the exponent. In other words, x(5) is the input vector at index 5, not x raised to the power of 5.
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6
Q

What is this?

|C|

A
  • Determinant if C is a matrix
  • Cardinality (number of member elements) if C is a set.
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7
Q

What’s a column vector?

A

Each entry occupies one row.

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8
Q

What’s this?

A

One use: denotes the i-th entry of a column vector

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9
Q

What does this mean?

A

Row vector. Each entry covers 1 column

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10
Q

For matrix A, what is

A

denotes the i-th row of A

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11
Q

For matrix A, what is

A

denotes the i-th column of A

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12
Q

What can dot product be applied to? What’s the formula?

A

Two vectors of same dimensions

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13
Q

What’s the inner product?

A

Same as dot product

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14
Q
  • What can outer product be applied to?
  • What does it result in?
  • What’s the formula?
A
  • Vectors (can have different number of elements)
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15
Q

What’s another way to express a dot product for two column vectors x and y?

A
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16
Q

What’s this for two vectors x and y?

A

The outer product

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17
Q

What’s a matrix-vector product?

A
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18
Q
  • What are the properties of matrix multiplication? (2)
  • What properties does it not have? (1)
A

Matrix multiplication is:

  • associative, (AB)C=A(BC)
  • distributive, A(B+C)=AB+AC

Is not:

  • commutative, AB≠BA
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19
Q

What’s a property of transposes?

A
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20
Q

What’s the Hadamard product?

A

elementwise matrix multiplication

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21
Q

What does this mean?

A

A standard basis vector - ith entry is 1 and the other entries are all 0

22
Q

What does this denote?

A

Identity matrix of nxn size

23
Q

What’s the point of a basis vector?

A

By multiplying a basis vector with a vector x∈Rn, we obtain the single entry xi

24
Q

Describe the idea of one-hot encoding

A
  • In each row of the resulting one-hot-encoded matrix:
  • In each row of the original vector, find the value
    • This value will serve as the column index of the “1” entry in the same row in the corresponding matrix.
    • The remaining values in that row are 0
25
Q

What’s a symmetric matrix?

A
26
Q

What’s this? What can it be applied to?

A
  • Inverse matrix
  • Only applied to square matrices
27
Q

What’s the fundamental rule of matrix inverses?

A
28
Q

What are the properties of matrix inverses?

A
29
Q

What’s a diagonal matrix?

A

a rectangular matrix with non-zero entries only on the main diagonal

30
Q

What’s this?

A
  • The p-norm (a vector norm).
  • It’s indicative of its magnitude or length
31
Q

What’s the formula of this?

A
32
Q

What’s this called? (2)

A
  • L2 norm (aka Euclidean norm):
33
Q

What’s this?

A

Gives the number of non-zero entries in the vector

34
Q

What’s this?

A

gives the maximum magnitude across the entries in vector x

35
Q

What’s this? What can it be applied to?

trace(A)

A
  • Sum of the elements across the main diagonal
  • Only applied to square matrices
36
Q
  • What’s this?
  • What can it be applied to?
A
  • quadratic form
  • See screenshot for applied to
37
Q

What’s vectorization?

A

a style of programming where operations are applied to whole arrays instead of individual elements (i.e. operate on entire data structure instead of loops)

38
Q

When should we use vectorization?

A

Whenever possible

39
Q

Why are python loops slower than C or Java loops?

A
  • Python performs variable type checks at each loop iteration
40
Q
  • What’s ndarray?
  • What’s a characteristic of its data structure?
A
  • Data structure with an interface similar to python lists, but its operations are performed in optimized C code
  • A homogeneous data structure
41
Q

What’s the runtime complexity of matrix multiplication using the original definition of MM? (optional)

A

O(n^3)

Description:

Here each entry in AB is the result of the dot product between a row in A and a column in B, which are both n-dimensional vectors. Therefore, it takes roughly n computations to yield one entry in AB, and because there are n2 entries, a total of n3 computations are required.

More formally, we say that the runtime complexity of matrix multiplication is O(n3). This notation indicates the growth rate of matrix multiplication – if we increase the size of A and B by a factor of 10, we will need (10n)3=1000n3 computations

42
Q

What do we know about the matrix multiplication algorithms of data science libraries?

A

They use algorithms that are more efficient than the textbook definition of matrix multiplication

43
Q
  • What’s a condition for vectorization?
  • What do we have to do if it’s not met?
  • What are 3 scenarios that pose this problem?
A
  • the elementwise operations are independent of each other
  • If not met, need to revert to using loops
  • Problem scenarios (examples in screenshot)
    • Loop dependency
    • Indirect memory access
    • Code branching
44
Q

How do you know if a function is vectorizable?

A

It doesn’t belong to the 3 problem scenarios of the independence condition

45
Q

How do you vectorize a function using numpy?

A

“register” it through the np.vectorize function.

46
Q

How does the performance of a custom vectorized function compare to alternatives?

A

Not as fast as functions of same functionality from DS libraries like numpy because they’re not executed in optimized C code

47
Q

How do you vectorize an interative function? (4)

A
  1. Given an iterative formula, organize its expected outputs in a suitable vector or matrix format.
  2. Break down this vector or matrix into smaller components.
  3. Vectorize each component by using matrix and vector operations. Honorable mentions include one-hot encoding, quadratic form, XXT along with XTX, and elementwise operators.
  4. Identify the appropriate NumPy functions to convert the vectorized expression into code.
48
Q

With two arrays x and y of the same size, how do you make a new array with some elements from x and some from y based on a condition?

A

numpy.where(condition, x, y)

where x and y are arrays

Returns an array with elements from x where the condition is True, and elements from y elsewhere.

49
Q

What are honorable mentions changes for when you can’t fully vectorize a component? (4)

A
  • one-hot encoding
  • quadratic form
  • XXT along with XTX
  • elementwise operators
50
Q

What should you be thinking when you’re trying to vectorized a function and you see this?

A

This is the same as a one-hot encoding then summing across the rows

51
Q

What’s a property of the binary indicator function?

A

𝟙(𝑎 & 𝑏)=𝟙(𝑎)×𝟙(𝑏).

52
Q

What should you think when you see a vector of dot products?

A

it very likely comes from a matrix-vector multiplication.

11637 Foundations of Computational Data Science (14 decks)

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